Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.448 - 0.893i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.298 − 0.0799i)2-s + (−1.64 + 0.540i)3-s + (−1.64 + 0.952i)4-s + (−1.56 + 1.59i)5-s + (−0.447 + 0.292i)6-s + (0.951 + 2.46i)7-s + (−0.852 + 0.852i)8-s + (2.41 − 1.77i)9-s + (−0.340 + 0.600i)10-s + (0.660 − 0.381i)11-s + (2.19 − 2.45i)12-s + (−2.27 − 2.27i)13-s + (0.481 + 0.660i)14-s + (1.71 − 3.47i)15-s + (1.71 − 2.97i)16-s + (−1.25 + 4.69i)17-s + ⋯
L(s)  = 1  + (0.210 − 0.0565i)2-s + (−0.950 + 0.312i)3-s + (−0.824 + 0.476i)4-s + (−0.701 + 0.712i)5-s + (−0.182 + 0.119i)6-s + (0.359 + 0.933i)7-s + (−0.301 + 0.301i)8-s + (0.805 − 0.593i)9-s + (−0.107 + 0.189i)10-s + (0.199 − 0.114i)11-s + (0.634 − 0.709i)12-s + (−0.629 − 0.629i)13-s + (0.128 + 0.176i)14-s + (0.443 − 0.896i)15-s + (0.429 − 0.744i)16-s + (−0.305 + 1.13i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.448 - 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.448 - 0.893i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (23, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ -0.448 - 0.893i)$
$L(1)$  $\approx$  $0.299270 + 0.485100i$
$L(\frac12)$  $\approx$  $0.299270 + 0.485100i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + (1.64 - 0.540i)T \)
5 \( 1 + (1.56 - 1.59i)T \)
7 \( 1 + (-0.951 - 2.46i)T \)
good2 \( 1 + (-0.298 + 0.0799i)T + (1.73 - i)T^{2} \)
11 \( 1 + (-0.660 + 0.381i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.27 + 2.27i)T + 13iT^{2} \)
17 \( 1 + (1.25 - 4.69i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.41 - 0.818i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.98 - 7.39i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 4.94T + 29T^{2} \)
31 \( 1 + (-2.96 - 5.13i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.915 + 3.41i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 4.35iT - 41T^{2} \)
43 \( 1 + (-2.69 - 2.69i)T + 43iT^{2} \)
47 \( 1 + (-4.14 + 1.10i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (6.71 + 1.79i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.84 + 6.65i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.19 - 3.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.0471 + 0.0126i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 12.4iT - 71T^{2} \)
73 \( 1 + (-0.359 + 1.34i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (3.66 + 2.11i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.05 + 5.05i)T - 83iT^{2} \)
89 \( 1 + (-0.453 + 0.785i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.73 + 3.73i)T - 97iT^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−14.21692182415470514151198545403, −12.69692785253600032597887368501, −12.06836799151734297787597491878, −11.16830411001891080312284504543, −9.986606306842582156626500887643, −8.676045940345749969920802671268, −7.47364299072514358021113402678, −5.89695763504139320325289300658, −4.74820219836353341832464297619, −3.38682358923095611902052775664, 0.73766198130993833711533773375, 4.47449016277750026230851239256, 4.83167292234714006105057090045, 6.57780663154673094210617388136, 7.75142580882094483501216726538, 9.176378494996287975274948437329, 10.34240648415674961191168762640, 11.49891927439940059660179153478, 12.37890719898763375871750840940, 13.39693421904091737788949744373

Graph of the $Z$-function along the critical line